Georgia Institute of Technology School of Mathematics | Georgia Institute of Technology | Atlanta, GA

We investigate dynamical systems obtained by coupling  an Anosov diffeomorphism and a N-pole-to-S-pole map of the circle. Both maps are uniformly hyperbolic; however, they have contrasting character, as the first one is chaotic while the second one has “orderly" dynamics. The first thing we show is that even weak coupling can produce interesting phenomena: when the attractor of the uncoupled system is not normally hyperbolic, most small interactions transform it from a smooth surface to a fractal-like set.  We then consider stronger couplings in which the action of the Anosov diffeomorphism on the circle map has certain monotonicity properties. These couplings produce genuine obstructions to uniform hyperbolicity; however, the monotonicity conditions make the system amenable to study by leveraging  techniques from the geometric and ergodic theories of hyperbolic systems.  In particular, we can show existence of invariant cones and SRB measures. 

This is joint work with Lai-Sang Young.

We provide an accurate description of the long time dynamics for generalized Korteweg-de Vries  and Benjamin-Ono equations on the circle without external parameters and for almost any (in probability and in density) small initial datum. To obtain that result we construct for these two classes of equations and under a very weak hypothesis of non degeneracy of the nonlinearity, rational normal forms on open sets surrounding the origin in high Sobolev regularity. With this new tool we can make precise the long time dynamics of the respective flows. In particular we prove a long-time stability result in Sobolev norm: given a large constant M and a sufficiently small parameter ε, for generic initial datum u(0) of size ε, we control the Sobolev norm of the solution u(t) for time of order ε^{−M}. 

This thesis has four chapters. The first three concern the location of mass on spheres or projective space, to minimize energies. For the Columb potential on the unit sphere, this is a classical problem, related to arranging electrons to minimize their energy. Restricting our potentials to be polynomials in the squared distance between points, we show in the Chapter 1 that there exist discrete minimal energy distributions. In addition we pose a conjecture on discreteness of minimizers for another class of energies while showing these minimizers must have empty interior.

In Chapter 2, we discover that highly symmetric distributions of points minimize energies over probability measures for potentials which are completely monotonic up to some degree, guided by the work of H. Cohn and A. Kumar. We make conjectures about optima for a class of energies calculated by summing absolute values of inner products raised to a positive power. Through reformulation, these observations give rise to new mixed-volume inequalities and conjectures. Our numerical experiments also lead to discovery of a new highly symmetric complex projective design which we detail the construction for. In this chapter we also provide details on a computer assisted argument which shows optimality of the $600$-cell for such energies (via interval arithmetic).

In Chapter 3 we also investigate energies having minimizers with a small number of distinct inner products. We focus here on discrete energies, confirming that for small $p$ the repeated orthonormal basis minimizes the $\ell_p $-norm of the inner products out of all unit norm configurations. These results have analogs for simplices which we also prove. 

Finally, in Chapter 4 we show that real tight frames that generate lattices must be rational, and that the same holds for other vector systems with structured matrices of outer products. We describe a construction of lattices from distance transitive graphs which gives rise to strongly eutactic lattices. We discuss properties of this construction and also detail potential applications of lattices generated by incoherent systems of vectors.

Adaptive control problems arise in many engineering applications in which one needs to design feedback controllers that ensure tracking of desired reference trajectories while at the same time identify unknown parameters such as control gains. This talk will summarize the speaker's work on adaptive tracking and parameter identification, including an application to curve tracking problems in robotics. The talk will be understandable to those familiar with the basic theory of ordinary differential equations. No prerequisite background in systems and control will be needed to understand and appreciate this talk.